Let $\alpha$ and $\beta$ be the roots of $x^2 - 6x - 2 = 0$,where $\alpha > \beta$. If $a_n = \alpha^n - \beta^n$ for all $n \geq 1$,then what is the value of $\frac{a_{10} - 2a_8}{2a_9}$?

The cubic equation whose roots are the squares of the roots of the equation $12x^3-20x^2+x+3=0$ is

If $\alpha$ and $\beta$ are the roots of the equation $2x^2-4x+3=0$,then $\frac{2(\alpha^4+\beta^4)+3(\alpha^2+\beta^2)}{\alpha+\beta} = $

If the roots of the equation $\frac{x^2 - bx}{ax - c} = \frac{m - 1}{m + 1}$ are equal in magnitude but opposite in sign,then $m = \dots$

If $\alpha, \beta, \gamma, \delta$ are in geometric progression,where $\alpha, \beta$ are the roots of the equation $ax^2 + 2bx + c = 0$ and $\gamma, \delta$ are the roots of the equation $px^2 + 2qx + r = 0$,then:

If $\alpha, \beta, \gamma$ are roots of the equation $x^3 + qx - r = 0$,then find the equation whose roots are $\left( \beta \gamma + \frac{1}{\alpha} \right), \left( \gamma \alpha + \frac{1}{\beta} \right), \left( \alpha \beta + \frac{1}{\gamma} \right)$.